**Instructor: ** Igor
Pak, MS 6125, pak@math.

**Class Schedule: ** MWF 12:00-12:50, MS 7608.

**Domino tilings**- W.P. Thurston, Conway's tiling groups (1990); the original article by Thurston describing his approach.
- J.C. Fournier, Tiling pictures of the plane with dominoes (1997); concise description of the algorithm and complexity applications.
- T. Chaboud, Domino tiling in planar graphs with regular and bipartite dual (1996); a small but insightful well written generalization.

**Combinatorial group theory**- J.H. Conway, J.C. Lagarias, Tiling with polyominoes and combinatorial group theory (1990); the original article.
- W.P. Thurston, Groups, Tings and Finite State Automata (1989); unpublished lecture notes describing the connection and giving more context.

**Ribbon tilings of Young diagram shapes**- I. Pak, Ribbon tile invariants (2000); the original article.
- S. Fomin, D. Stanton Rim hook lattices (1997); a concise description of the rim hook bijection.

**Tile invariants and ribbon tilings of general regions**- C. Moore, I. Pak, Ribbon tile invariants from signed area (2002); the original article using C-L approach.
- R. Muchnik, I. Pak, On tilings by ribbon tetrominoes (1999); here Lemma 2.1 (the "induction lemma") is given with a proof which is omitted in C-L paper.
- M. Korn, Geometric and algebraic properties of polyomino tilings (MIT thesis, 2004); Section 10 gives a clean alternative proof, Section 12 gives a counterexample to the inductive lemma in 3-dim.
- S. Sheffield, Ribbon tilings and multidimensional height functions (2002); proof of 2-connectivity for ribbon tilings and Thurston style tileability algorithm.

**Rectangles with one side integral**- S. Wagon, Fourteen Proofs of a Result About Tiling a Rectangle (1988); original article with 14 proofs (not all of them clear or valid)
- D.J.C. MacKay, Simple Proofs of a Rectangle Tiling Theorem, useful variations and additional details on these proofs.
- P. Winkler, Integers and Rectangles,
in
*Mathematical Puzzles*(2004), another proof. - R. Kenyon, A note on tiling with integer-sided rectangles (1994); a new group theory style proof with algorithmic applications.
- N.G. de Bruijn, Filling Boxes with Bricks (1969), the original motivation behind this study.
- V.G. Pokrovskii, Linear relations in dissections into n-dimensional parallelepipeds (1985); an interesting but little cited generalization of de Bruijn's theorem (free access Russian original)

**Tilings with two bars**- C. Kenyon, R. Kenyon, Tiling a polygon with rectangles (1992); the original article.
- D. Beauquier, M. Nivat, E. Rémila, M. Robson, Tiling figures of the plane with two bars (1995); a complementary hardness result.

**T-tetromino tilings**- D.W. Walkup, Covering a rectangle with T-tetrominoes (1965), tileability of rectangles
- M. Korn, I. Pak, Tilings of rectangles with T-tetrominoes (2004); height function, local move connectivity and counting

**Further applications of height functions**- M. Luby, D. Randall, A. Sinclair, Markov chain algorithms for planar lattice structures (2001), height function for 3-colorings of the grid
- R. Kenyon, Tiling a polygon with parallelograms (1992); poly time algorithm.

**Tilings of rectangles**- N.G. de Bruijn, D.A. Klarner, A finite basis theorem for packing boxes with bricks (1975), original article (first correct proof in full generality)
- M. Reid, Klarner Systems and Tiling Boxes with Polyominoes (2004), a simplified proof with interesting applications
- J. Yang, Rectangular tileability and complementary tileability are undecidable (2012), a complementary hardness result.

**Tilings of rectangles with rectangles**- F.W. Barnes, Algebraic theory of brick packing, part I and part II (1982); original breakthrough article
- M. Reid, Asymptotically Optimal Box Packing Theorems (2008), a Klarner system style proof and some amazing examples
- T. Lam, Tiling with commutative rings (2008), an intro to commutative ring method
- O. Bodini, Tiling a Rectangle with Polyominoes (2003), an algebraic proof, barely readable.

**Order of tiles**- D.A. Klarner, Packing a rectangle with
congruent
*N*-ominoes (1969); a number of interesting examples. - M. Reid, Tiling rectangles and half strips with congruent polyominoes (1997); more examples and discussion of the odd order conjecture.

- D.A. Klarner, Packing a rectangle with
congruent
**Augmentability**- Yang's paper (see above); hardness of augmentability and decidability of augmentability with rectangles.
- Korn's thesis (see above), Section 11; negative solution of augmentability for dominoes.

**Valuations**- C. Freiling, D. Rinne, Tiling a square with similar rectangles (1994); necessary and sufficient conditions for tiling squares with rectangles similar to a given.
- M. Laczkovich, G. Szekeres, Tiling of the square with similar rectangles (1995); an independent proof.
- C. Freiling, M. Laczkovich, D. Rinne, Rectangling a rectangle (1997); a generalization to rectangular regions.
- M. Prasolov, M. Skopenkov, Tiling by rectangles and alternating current (2010); an alternative proof.

**Counting domino tilings and perfect matchings**- L. Lovász, M.D. Plummer,
*Matching Theory*, Chapter 8; among other things, general Pfaffian-Determinant approach. - R. Kenyon, An introduction to the dimer model (2002); variations on Kasteleyn method, isoradial graphs.
- R.W. Kenyon, J.G. Propp, D.B. Wilson, Trees and Matchings (2000); Temperley's bijection and applications.

- L. Lovász, M.D. Plummer,
**Aztec diamond**- A. Björner, R.P. Stanley, A combinatorial miscellany (2010); domino shuffling.
- N. Elkies, G. Kuperberg, M. Larsen, J. Propp, Alternating-sign matrices and domino tilings; the original article with four proofs.
- Jim Propp, San Diego talk slides, outline of a short proof based on the "urban renewal".
- S.-P. Eu and T.-S. Fu, A simple proof of the Aztec diamond theorem (2005), an elegant short proof based on the non-intersecting paths principle.
- K.P. Kokhas, Domino tilings of aztec diamonds and squares (2008); yet another beautiful proof of the Aztec diamond formula via a general recurrence relation (§ 2.1 and 2.2)
- L. Pachter, Combinatorial Approaches and Conjectures for 2-Divisibility Problems Concerning Domino Tilings of Polyominoes (1997); simple proof of divisibility of the number of domino tilings of a even square.

*Warnings:* Most of these links are external, some are by subscription, some can be broken; occasionally, their content is unverified.
Also, the explanations are NOT review, but rather quick summary of material I used from the sources; often there is wealth of other work
presented there as well.

- Federico Ardila and Richard P. Stanley, Tilings (2005), an introductory article.
- Branko Grunbaum and G. C. Shephard,
*Tilings and Patterns*(1986), an instant classic. A generic introduction to the subject. - Solomon W. Golomb,
*Polyominoes: Puzzles, Patterns, Problems, and Packings*, true classic in the recreational literature (Second Ed., 1995).

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*Last updated 3/8/2013*